Dr. Loveless Curiosity Lab

Building Shells with Calculus III

Polar Spirals, Nautilus Growth, Moving Frames, and 3D Shell Models

We started with a simple question: how can we make seashells in Desmos? Shells have interesting geometry: curves, spirals, surfaces, chambers, openings, and growth patterns.

After a short literature review, we realized that shell models often begin with two major mathematical tools: logarithmic spirals, which describe self-similar growth, and the TNB frame, which gives a way to move an opening through space. This project became a visual exploration of how those Calculus III tools can build shell-like surfaces.

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Explore Shell Growth

Explore how shell models grow from simple spirals into full 3D surfaces: start with polar growth, fit a nautilus-style spiral, build a moving aperture with the TNB frame, and then sweep that aperture into interactive shell models.

Concept 1: Polar Curves and Spirals
Spiral growth examples: constant, square-root, and exponential growth in polar coordinates.

Pick a spiral type. The picture mode shows a nature-inspired overlay; interactive mode removes the background and turns on axes so you can adjust the parameters.

Spiral growth examples
Growth 100%
Loading polar spiral visual...
Concept 2: Nautilus Shell Growth in 2D
Use logarithmic spirals to trace a nautilus-style shell and compare the fitted growth rate with the golden spiral.

Shell spirals have a special growth property: the log of the ratio of two radii is proportional to the change in angle. This defines a logarithmic spiral.

For nautilus shells, the proportionality constant is often reported around 0.185. Many websites call the nautilus a “golden spiral.” Both curves are logarithmic, but the golden spiral has a larger growth rate, so it is not quite as good a fit.

In this visual, we match a spiral to a nautilus photo and check that the growth constant is roughly what appears in research articles. Click Golden Spiral to compare the faster golden-ratio growth rate.

Nautilus spiral model
Growth time \(t\) t = 0.00
Loading shell growth visual...
Concept 3: Building Tubes Around Curves
Use the TNB frame to attach a moving aperture to a curve, then sweep that aperture into a tube-like surface.

The TNB frame is a powerful Calculus III tool for analyzing curves, but the hand calculations can be tedious. Here, Desmos computes the derivatives so we can focus on what the vectors are good for.

In the plane, a circle around a center \(C\) can be written as \(C+r\cos(\theta)(1,0)+r\sin(\theta)(0,1)\). In 3D, the TNB frame gives us new coordinate directions: the normal vector \(N\) and binormal vector \(B\). So a circle around a space curve becomes \(C+r\cos(\theta)N+r\sin(\theta)B\).

Press play to move along the curve, reveal the TNB frame, show the normal plane, then add the aperture circle. Last, sweep the aperture to build a tube around the curve.

Build a tube with the TNB frame
Preset curve
Parameter \(t\) t = 0.00
Aperture radius r = 0.30
Loading moving-frame visual...
Concept 4: Shell Models in 3D
Toggle between a chambered nautilus model and a tall spiral shell model. Press play to watch each shell grow.

A shell can be modeled as a growing aperture swept along a logarithmic spiral path. The final models combine the earlier ideas: spiral growth gives the path, and a moving frame tells the opening how to travel through space.

The chambered model adds walls between turns of a nautilus-style spiral. The tall model sweeps a growing aperture upward to create a cone-like spiral shell. These are simplified mathematical models, but they show how curves, frames, and surfaces work together.

Choose a shell, press play to watch it grow, then gently adjust the parameters to see how growth, spacing, aperture size, ribs, and spines change the surface.

Choose a 3D shell model
Growth growth = 1.00
Loading 3D shell models...

Where It Started

This project started with the question: how can we make seashells in Desmos? That question quickly led to polar spirals, surfaces, and moving frames.

The result is a visual path through two Calculus III ideas that can feel abstract at first, but become useful when we use them to build natural forms.

Going Further

Future versions could compare fitted shell parameters across species, adjust the aperture from a circle to an ellipse, and explore which features require a true 3D surface rather than a 2D spiral.

Other natural extensions include sliders for ribs and spines, a small gallery of shell presets, and a student-written one-page summary explaining the model and literature.

This connects naturally to polar curves, parametric curves, parametric surfaces, curvature, torsion, moving frames, and growth models.

Further Reading